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This book focuses on linear time eigenvalue location algorithms for
graphs. This subject relates to spectral graph theory, a field that
combines tools and concepts of linear algebra and combinatorics,
with applications ranging from image processing and data analysis
to molecular descriptors and random walks. It has attracted a lot
of attention and has since emerged as an area on its own. Studies
in spectral graph theory seek to determine properties of a graph
through matrices associated with it. It turns out that eigenvalues
and eigenvectors have surprisingly many connections with the
structure of a graph. This book approaches this subject under the
perspective of eigenvalue location algorithms. These are algorithms
that, given a symmetric graph matrix M and a real interval I,
return the number of eigenvalues of M that lie in I. Since the
algorithms described here are typically very fast, they allow one
to quickly approximate the value of any eigenvalue, which is a
basic step in most applications of spectral graph theory. Moreover,
these algorithms are convenient theoretical tools for proving
bounds on eigenvalues and their multiplicities, which was quite
useful to solve longstanding open problems in the area. This book
brings these algorithms together, revealing how similar they are in
spirit, and presents some of their main applications. This work can
be of special interest to graduate students and researchers in
spectral graph theory, and to any mathematician who wishes to know
more about eigenvalues associated with graphs. It can also serve as
a compact textbook for short courses on the topic.
Nathaniel J. Cauliflower's distinguished afterlife role as Musing
Department caretaker is thrust into detrimental jeopardy when a
single inspiration is neglected and absconded by one of his muses,
Jonas Rothchild. With the very nature of time fracturing apart,
Nathaniel enlists the help of the division's nine greatest modern
day muses to assist in its provisional repair. Among these
encouraging leaders is previous antisocial motivator Annette
Slocum. Much to Nathaniel's chagrin, Annette's presence in the
office stirs a ferocious unraveling of his secret backstory. His
unfolding is an epic tale of deceit and murder that sweeps from
19th century Paris to modern day, giving rise to an underlying
malevolent agenda atop an expansive field of dandelions where three
fateful, interweaving fortune-telling Sisters reside. When it is
discovered that Jonas is stealing other inspirations, with the
anonymous assistance from one of the department's own, Nathaniel's
unsettling past vicariously forces him into the spotlight as the
prime suspect. With time splintering further, Nathaniel must clear
his name by taking ownership of his previous, darkest,
soul-shattering affairs and fight to reclaim the hereafter agency
that he, for so many centuries, has devotedly sought to protect.
Annette Slocum's tragically boring unfulfilled life truly begins
when she is struck by a Cadillac in front of her local library. In
the afterlife Annette is indoctrinated as a modern day muse.
Annette travels backwards and forwards through time, by use of a
Hasbro Lite-Brite, providing catalysts of inspiration to the
living. Unfortunately for Annette, one single misstep in the
proceedings could cause a chain reaction that might decimate the
already fragile tapestry of the Universe. Severely antisocial,
Annette must now commune with the world that terrifies her. She
must overcome her own inner demons and personal insecurities if she
is to effectively inspire. It doesn't help matters when Annette's
childhood foe, now adult adversary, Jonas Rothchild, resurfaces in
the life of her favorite client. Jonas' past involvement in
tormenting her escalates to dangerous levels, forcing Annette to
find the bravery to stand up to her infamous villain once and for
all. Rhapsody is a novel about the living and the muses who inspire
them, treacherous violet envelopes, the insurmountable search for
strength and the enthralling quest to uncover the secrets of an
elusive stranger named Nathaniel J. Cauliflower who holds the key
to the entire operation.
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